Deck 3: Motion in Two Dimensions

Question 1

A space station of diameter 80 m is turning about its axis at a constant rate. If the acceleration of the outer rim of the station is 2.5 m/s2, what is the period of revolution of the space station?

A)22 s
B)19 s
C)25 s
D)28 s
E)40 s

Answer 1

25 s

Question 2

An object moving at a constant speed requires 6.0 s to go once around a circle with a diameter of 4.0 m. What is the magnitude of the instantaneous acceleration of the particle during this time?

A)2.2 m/s2
B)2.7 m/s2
C)3.3 m/s2
D)3.8 m/s2
E)4.4 m/s2

Answer 2

2.2 m/s2

Question 3

There are floods in outback Queensland and you are on a mission to deliver supplies to stranded households. You plan to drop them from an aircraft which will be flying horizontally with a speed of 300 m/s at an altitude of 400 m. Assume that the ground is level. At what horizontal distance from a household must the pilot drop the supplies so that they reach the household?

A)3.0 km
B)2.4 km
C)3.3 km
D)2.7 km
E)1.7 km

Answer 3

2.7 km

Question 4

A rock is projected from the edge of the top of a building with an initial velocity of 12.2 m/s at an angle of 53 $\degree$ above the horizontal. The rock strikes the ground a horizontal distance of 25 m from the base of the building. Assume that the ground is level and that the side of the building is vertical. How tall is the building?

A)25.3 m
B)29.6 m
C)27.4 m
D)23.6 m
E)18.9 m

Question 5

A cricket ball is hit at ground level. The ball is observed to reach its maximum height above ground level 3.0 s after being hit. And 2.5 s after reaching this maximum height, the ball is observed to barely clear a fence that is 97.5 m from where it was hit. How high is the fence?

A)8.2 m
B)15.8 m
C)13.5 m
D)11.0 m
E)4.9 m

Question 6

A car travels counterclockwise around a flat circle of radius 0.25 km at a constant speed of 20 m/s. When the car is at point A as shown in the figure, what is the car's acceleration?

A)1.6 m/s2, south
B)zero
C)1.6 m/s2, east
D)1.6 m/s2, north
E)1.6 m/s2, west

Question 7

A 0.20-km wide river has a uniform flow speed of 4.0 m/s toward the east. It takes 20 s for a boat to cross the river to a point directly north of its departure point on the south bank. In what direction must the boat be pointed in order to accomplish this?

A)23 $\degree$ west of north
B)20 $\degree$ west of north
C)24 $\degree$ west of north
D)22 $\degree$ west of north
E)17 $\degree$ west of north

Question 8

A stunt pilot performs a circular dive of radius 800 m. At the bottom of the dive (point B in the figure) the pilot has a speed of 200 m/s which at that instant is increasing at a rate of 20 m/s2. What acceleration does the pilot have at point B?

A)(50i + 20j) m/s2
B)(20i -50j) m/s2
C)(20i + 50j) m/s2
D)(-20i + 50j) m/s2
E)(-50i + 20j) m/s2

Question 9

The pilot of an aircraft flies due north relative to the ground in a wind blowing 40 km/h toward the east. If his speed relative to the ground is 80 km/h, what is the speed of his airplane relative to the air?

A)89 km/h
B)85 km/h
C)81 km/h
D)76 km/h
E)72 km/h

Question 10

A car travels in a flat circle of radius R. At a certain instant the velocity of the car is 24 m/s west, and the acceleration of the car has components of 2.4 m/s2 east and 1.8 m/s2 south. What is the radius of the circle?

A)0.24 km
B)0.19 km
C)0.32 km
D)0.14 km
E)0.27 km

Question 11

A particle leaves the origin with a velocity of 7.2 m/s in the positive y direction and moves in the xy plane with a constant acceleration of (3.0 $\hat{\mathbf{i}}$ - 2.0 $\hat{\mathbf{j}}$ ) m/s2. At the instant the particle moves back across the x axis (y = 0), what is the value of its x coordinate?

A)65 m
B)91 m
C)54 m
D)78 m
E)86 m

Question 12

A racing car moving with a constant speed of 60 m/s completes one lap around a circular track in 50 s. What is the magnitude of the acceleration of the race car?

A)8.8 m/s2
B)7.5 m/s2
C)9.4 m/s2
D)6.3 m/s2
E)5.3 m/s2

Question 13

A car travels in a flat circle of radius R. At a certain instant the velocity of the car is 20 m/s north, and the total acceleration of the car is 2.5 m/s2 37 $\degree$ south of west. Which of the following is correct?

A)R = 0.40 km, and the car's speed is decreasing.
B)R = 0.20 km, and the car's speed is decreasing.
C)R = 0.20 km, and the car's speed is increasing.
D)R = 0.16 km, and the car's speed is increasing.
E)R = 0.16 km, and the car's speed is decreasing.

Question 14

A Ferris wheel at Luna Park has a 15-m radius and completes five turns about its horizontal axis every minute. What is the acceleration of a passenger at his lowest point during the ride?

A)5.7 m/s2 downward
B)4.1 m/s2 upward
C)14 m/s2 downward
D)4.1 m/s2 downward
E)19 m/s2 downward

Question 15

A 0.14-km wide river flows with a uniform speed of 4.0 m/s toward the east. It takes 20 s for a boat to cross the river to a point directly north of its departure point on the south bank. What is the speed of the boat relative to the water?

A)5.7 m/s
B)8.5 m/s
C)8.1 m/s
D)7.0 m/s
E)6.4 m/s

Question 16

Starting from one oasis, a camel walks 25 km in a direction 30 $\degree$ south of west and then walks 30 km toward the north to a second oasis. What distance separates the two oases?

A)15 km
B)48 km
C)28 km
D)53 km
E)55 km

Question 17

At t = 0, a particle leaves the origin with a velocity of 9.0 m/s in the positive y direction and moves in the xy plane with a constant acceleration of (2.0i - 4.0j) m/s2. At the instant the x coordinate of the particle is 15 m, what is the speed of the particle?

A)10 m/s
B)16 m/s
C)12 m/s
D)14 m/s
E)26 m/s

Question 18

The initial speed of a cannon ball is 0.20 km/s. If the ball is to strike a target that is at a horizontal distance of 3.0 km from the cannon, what is the minimum time of flight for the ball?

A)16 s
B)21 s
C)24 s
D)14 s
E)19 s

Question 19

A car travels in a due northerly direction at a speed of 55 km/h. The traces of rain on the side windows of the car make an angle of 60 degrees with respect to the horizontal. If the rain is falling vertically with respect to the earth, what is the speed of the rain with respect to the earth?

A)48 km/h
B)95 km/h
C)58 km/h
D)32 km/h
E)80 km/h

Question 20

A rock is thrown from the edge of the top of a 100-ft tall building at some unknown angle above the horizontal. The rock strikes the ground a horizontal distance of 160 ft from the base of the building 5.0 s after being thrown. Assume that the ground is level and that the side of the building is vertical. Determine the speed with which the rock was thrown.

A)72 ft/s
B)77 ft/s
C)68 ft/s
D)82 ft/s
E)87 ft/s

Question 21

A tennis player at the Australian Open wants to slam a serve at O so that the ball lands just inside the opposite corner of the court. What should the ratio $\frac{v_{0 y}}{v_{0 x}}$ be for the initial velocity $\overrightarrow{\mathbf{v}}_{0}$ ? The time t = 0 is the time when the ball is hit by the racket.

A)W/L
B)L/W
C) $\frac{1}{2} g t^{2} / L$
D) $\frac{1}{2} g t^{2} / W$
E) $\frac{1}{2} g t^{2} / \sqrt{L^{2}+W^{2}}$

Question 22

The vector $\overrightarrow{\mathbf{r}}$ indicates the instantaneous displacement of a projectile from the origin. At the instant when the projectile is at $\overrightarrow{\mathbf{r}}$ , its velocity and acceleration vectors are $\overrightarrow{\mathbf{v}}$ and $\overrightarrow{\mathbf{a}}$ . Which statement is correct?

A) $\overrightarrow{\mathbf{v}}$ is always perpendicular to $\overrightarrow{\mathbf{r}}$ .
B) $\overrightarrow{\mathbf{a}}$ is always perpendicular to $\overrightarrow{\mathbf{r}}$ .
C) $\overrightarrow{\mathbf{a}}$ is always perpendicular to $\overrightarrow{\mathbf{v}}$ .
D) $\overrightarrow{\mathbf{a}}$ is always perpendicular to $\overrightarrow{\mathbf{V}}_{x}$ .
E) $\overrightarrow{\mathbf{a}}$ is always perpendicular to $\overrightarrow{\mathbf{v}}_{y}$ .

Question 23

The site from which an airplane takes off is the origin. The x axis points east; the y axis points straight up. The position and velocity vectors of the plane at a later time are given by: $\overrightarrow{\mathbf{r}}=\left(1.61 \times 10^{6} \hat{\mathbf{i}}\right) \mathrm{m}$ and $\overrightarrow{\mathbf{v}}=+100 \hat{\mathbf{i}} \frac{\mathrm{~m}}{\mathrm{~s}}$ .The plane is most likely:

A)just touching down.
B)in level flight in the air.
C)ascending.
D)descending.
E)taking off.

Question 24

A Olympic athlete in the long jump goes into the jump at 12 m/s and launches himself at 20 $\degree$ above the horizontal. How long is he in the air before returning to Earth?

Question 25

A car is driven 1200 m north at 20.0 m/s and then driven 1600 m east at 25.0 m/s. What is the magnitude of the average velocity for this trip?

A)16.1 m/s
B)22.6 m/s
C)31.3 m/s
D)11.3 m/s
E)62.2 m/s

Question 26

The site from which an airplane takes off is the origin. The x axis points east; the y axis points straight up. The position and velocity vectors of the plane at a later time are given by: $\overrightarrow{\mathbf{r}}=\left(1.61 \times 10^{6} \tilde{\mathbf{i}}+3.00 \times 10^{3} \hat{\mathbf{j}}\right) \mathrm{m}$ and $\overrightarrow{\mathbf{v}}=(150 \hat{\mathbf{i}}+21 \hat{\mathbf{j}}) \frac{\mathrm{m}}{\mathrm{~s}}$ .The plane is most likely:

A)just touching down.
B)in level flight in the air.
C)ascending.
D)descending.
E)taking off.

Question 27

The position of an object is given by $\overrightarrow{\mathbf{r}}=\left(-4.00 t \hat{\mathbf{i}}+6.00 t^{3} \hat{\mathbf{j}}\right) \mathrm{m}$ where t is in seconds. At t = 2.0 s, what is the magnitude of the particle's acceleration?

A)0 m/s2
B)2.0 m/s2
C)17 m/s2
D)36 m/s2
E)72 m/s2

Question 28

In a location where the train tracks run parallel to a road, a sports car travelling at 60 m/s passes a freight train travelling at 30 m/s. How long does it take for the car to be 180 m ahead of the train?

A)2.0 s
B)3.0 s
C)6.0 s
D)9.0 s
E)18.0 s

Question 29

A car travels around an oval racetrack at constant speed. The car is accelerating

A)at all points except B and D.
B)at all points except A and C.
C)at all points except A, B, C, and D.
D)everywhere, including points A, B, C, and D.
E)nowhere, because it is travelling at constant speed.

Question 30

A block is supported on a compressed spring, which projects the block straight up in the air at velocity $\overrightarrow{\mathbf{v}}=v_{0y} \hat{\mathbf{j}}$ . The spring and ledge it sits on then retract. You can win a prize by hitting the block with a ball. When should you throw the ball and in what direction to be sure the ball hits the block? (Assume the ball can reach the block before the block reaches the ground and that the ball is thrown from a height equal to the release position of the block.)

A)At the instant when the block leaves the spring, directed at the block.
B)At the instant when the block leaves the spring, directed at the spring.
C)At the instant when the block is at the highest point, directed at the block.
D)At the instant when the block is at the highest point, directed at the spring.
E)When the block is back at the spring's original position, directed at that position.

Question 31

Two cars are travelling around identical circular racetracks. Car A travels at a constant speed of 20 m/s. Car B starts at rest and speeds up with constant tangential acceleration until its speed is 40 m/s. When car B has the same (tangential) velocity as car A, it is always true that:

A)it is passing car A.
B)it has the same linear (tangential) acceleration as car A.
C)it has the same centripetal acceleration as car A.
D)it has the same total acceleration as car A.
E)it has travelled farther than car A since starting.

Question 32

A car is driven 1200 m north at 20.0 m/s and then driven 1600 m east at 25.0 m/s. What are the magnitude and direction of the displacement for this trip?

A)1400 m, northeast
B)2000 m, 36.9° north of east
C)2000 m, 53.1° north of east
D)2800 m, 36.9° east of north
E)2800 m, 53.1° east of north

Question 33

Two vectors starting at the same origin have equal and opposite x components. Is it possible for the two vectors to be perpendicular to each other? Justify your answer.

Question 34

A vector starts at coordinate (3.0, 4.0) and ends at coordinate (-2.0, 16.0). What are the magnitude and direction of this vector?

Question 35

A projectile starts at the coordinate origin, where the displacement vector also originates. The initial velocity, v0, makes an angle $\theta$ 0 with the horizontal where 0 < $\theta$ 0 < 90 $\degree$ . At the instant when the projectile is at the highest point of its trajectory, the displacement, velocity and acceleration vectors are $\overrightarrow{\mathbf{r}}$ , $\overrightarrow{\mathbf{v}}$ and $\overrightarrow{\mathbf{a}}$ . Which statement is true?

A) $\overrightarrow{\mathbf{r}}$ is parallel to $\overrightarrow{\mathbf{v}}$ .
B) $\overrightarrow{\mathbf{r}}$ is perpendicular to $\overrightarrow{\mathbf{v}}$ .
C) $\overrightarrow{\mathbf{v}}$ is parallel to $\overrightarrow{\mathbf{a}}$ .
D) $\overrightarrow{\mathbf{v}}$ is perpendicular to $\overrightarrow{\mathbf{a}}$ .
E) $\overrightarrow{\mathbf{r}}$ is perpendicular to $\overrightarrow{\mathbf{a}}$ .

Question 36

Jane plans to fly from Canberra to Adelaide, about 1000 km due west of Canberra. She heads due west at 100 km/h for 10 hours but finds herself at Broken Hill, which is 370 km from Adelaide in a direction 45 degrees north of due east. What was the wind velocity?

A)37 km/h, E
B)37 km/h, N
C)37 km/h, NE
D)74 km/h, N
E)74 km/h, NE

Question 37

Two balls, projected at different times so they don't collide, have trajectories A and B, as shown below. Which statement is correct?

A)v0B must be greater than v0A.
B)Ball A is in the air for a longer time than ball B.
C)Ball B is in the air for a longer time than ball A.
D)Ball B has a greater acceleration than ball A.
E)Ball A has a greater acceleration than ball B.

Question 38

Imagine that you are going for a ride in a hot air balloon at the Canowindra Balloon Challenge. While the balloon is rising at a speed of 2.0 m/s, you throw a small ball down at a speed of 5.0 m/s relative to your body. A person who measures the ball's velocity at the instant of release will find that the ball's velocity relative to the ground at that instant is

A)2.0 m/s, up.
B)3.0 m/s, down.
C)3.0 m/s, up.
D)5.0 m/s, down.
E)12.8 m/s, down.

Question 39

A student in the front of a school bus tosses a ball to another student in the back of the bus while the bus is moving forward at constant velocity. The speed of the ball as seen by a stationary observer in the street:

A)is less than that observed inside the bus.
B)is the same as that observed inside the bus.
C)is greater than that observed inside the bus.
D)may be either greater or smaller than that observed inside the bus.
E)may be either greater, smaller, or equal to that observed inside the bus.

Question 40

A motorcycle daredevil wants to ride up a 50.0 m ramp set at a 30.0 $\degree$ incline to the ground. It will launch him in the air and he wants to come down so he just misses the last of a number of 1.00 m diameter barrels. If the speed at the instant when he leaves the ramp is 60.0 m/s, how many barrels can be used?

A)79
B)318
C)332
D)355
E)402

Question 41

A tennis player standing 12.6 m from the net hits the ball at 3.00 $\degree$ above the horizontal. To clear the net, the ball must rise at least 0.330 m. If the ball just clears the net at the apex of its trajectory, how fast was the ball moving when it left the racket?

Question 42

A satellite is in a circular orbit 600 km above the Earth's surface. The acceleration of gravity is 8.21 m/s2 at this altitude. The radius of the Earth is 6400 km. Determine the speed of the satellite, and the time to complete one orbit around the Earth.

Question 43

A football is kicked upward at a 30.0 $\degree$ angle to the horizontal. To kick a 40.0-metre pass, what must be the initial speed of the ball?

Question 44

Raindrops are falling straight downward. When observed from a car travelling at 55.0 mi/h, the drops streak the side window at an angle of 60.0 $\degree$ with the vertical. Find the speed with which the drops are falling.

Question 45

An explorer wishes to cross a river that is 1.5 km wide and flows with a velocity of 5.0 km/h parallel to its banks. The explorer uses a small powerboat that moves at a maximum speed of 12 km/h with respect to the water. What is the minimum time for crossing?